## Franklin's Notes

### Preorder

A preorder is a binary relation $R\subset A\times A$ on a set $A$ that is both reflexive and transitive. That is:

1. $aRa$ for all $a\in A$
2. If $aRb$ and $bRc$, then $aRc$, for all $a,b,c\in A$

The name "preorder" is apt because a preorder is essentially a poset in which some "equivalent" elements have not yet been "collapsed" into the same element. That is, a preorder may have elements $a,b$ such that $aRb$ and $bRa$ while $a\ne b$. This means that if an equivalence relation $\sim \subset A\times A$ is defined by then the preorder $R$ can be converted into a partial ordering $R'$ on the quotient set $A/\sim$ by letting $[a]R'[b]$ iff $aRb$.

Preorders can also be interpreted from a categorical viewpoint as a thin category, or a category with at most one morphism between any pair of elements.

# order-theory

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